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Estonian Mathematical Olympiad

Algebra Difficulty 5.0 AIME Find the answer Estonia

The letters $M$ , $A$ , $T$ , $E$ , $I$ and $K$ are assigned the numbers $1$ , $2$ , $3$ , $4$ , $5$ and $6$ in a certain order so that different letters correspond to different numbers. The sum of the numbers corresponding to the letters of the word $MATEMAATIK$ (taking into account repetitions) is $42$ and the sum of the numbers corresponding to the letters of the word $KEEMIK$ (taking into account repetitions) is $13$ . Find the sum of the numbers corresponding to the letters of the word $IT$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

In the words $MATEMAATIK$ , $KEEMIK$ and $IT$ , all the given letters occur exactly $3$ times. Since the sum of the numbers corresponding to the letters $M$ , $A$ , $T$ , $E$ , $I$ and $K$ is $1+2+3+4+5+6=21$ , the sum of the numbers corresponding to the letters in these three words is $3 \cdot 21 = 63$ . Consequently, the sum of the numbers corresponding to the letters in the word $IT$ is $63 - 42 - 13 = 8$ .

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